| dc.description.abstract | The localization of energy-momentum for gravitating systems has remained an
important problem since the time of Einstein. Due to the equivalence principle
there is no proper energy-momentum density. Traditional approaches led to a va-
riety of reference frame dependent expressions, referred to as pseudotensors. A
more modern idea is quasilocal energy-momentum. C.M. Chen, using a covariant
Hamiltonian formalism, gave four preferred Hamiltonian boundary term quasilocal
energy-momentum expressions. The classical pseudotenor expressions, as well as the
quasilocal expressions generally agree for the total (i.e. global) values but give quite
di®erent values locally. It is desirable to ¯nd some way to choose which expression
gives a better description of the energy-momentum for a gravitating system. Here
we shall test both the well-known classical pseudotensors (in particular, Einstein,
Papapetrou, Landau-Lifshits ’’ Bergmann-Thomson, M¿ller (1958), M¿ller (1961),
Weinberg) and the covariant Hamiltonian quasilocal boundary expressions in a dif-
ferent regime, namely the small region limit|both inside matter and in vacuum.
All of the expressions|except for M¿ller’’s 1958 expression|give the correct mate-
rial limit. In small vacuum regions we found some interesting results which allows
us to choose which expressions satisfy an important physical property: positive en-
ergy. None of the classical pseudotensors satis¯es this positivity property, however
there is a one-parameter set of linear combinations which, to lowest non-vanishing
order is proportional to the Bel-Robinson tensor and hence is positive for small
regions. Moreover, we have constructed an in¯nite set (with 10 constant parame-
ters) of additional new holonomic pseudotensors which, although rather contrived,
satisfy this important positive energy requirement. On the other hand we found
that M¿ller’’s 1961 teleparallel-tetrad energy-momentum expression naturally has
this Bel-Robinson property. For C.M. Chen’’s covariant-symplectic quasilocal ex-
pressions we found that one, corresponding to the natural boundary choices, gives
this desired Bel-Robinson positivity result in orthonormal frames. Moreover within
a two parameters modi¯cation of the Chen-Nester four expressions, one gives an
extra nice result in holonomic frames. | en_US |